Electronic Journal of Differential Equations, Vol. 2000(2000), No. 42, pp. 1–25.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login: ftp)
Spectrum of the linearized operator for the
Ginzburg-Landau equation ∗
Tai-Chia Lin
Abstract
We study the spectrum of the linearized operator for the GinzburgLandau equation about a symmetric vortex solution with degree one. We
show that the smallest eigenvalue of the linearized operator has multiplicity two, and then we describe its behavior as a small parameter approaches
zero. We also find a positive lower bound for all the other eigenvalues,
and find estimates of the first eigenfunction. Then using these results, we
give partial results on the dynamics of vortices in the nonlinear heat and
Schrödinger equations.
1
Introduction
We consider the steady state for the Ginzburg-Landau equation
∆u +
1
u(1 − |u|2 ) = 0
2
for x ∈ R2 ,
(1.1)
where the solution u is a complex-valued function and is a small positive parameter. Symmetric vortex solution to (1.1) with degree one has been obtained
in [3, 10, 11]. The solution have the form
r
u(x) = U1 (x) = f0 ( )eiθ ,
where (r, θ) denote the polar coordinates of x ∈ R2 and f0 = f0 (s) is the solution
of
0
−f 00 − fs + s12 f = f · (1 − f 2 ) for s > 0 ,
(1.2)
f (+∞) = 1, f (0) = 0, f ≥ 0 .
Moreover, f0 (s) satisfies
f0 (s)
= s(A0 +
∞
X
P2k (A0 )s2k ) for s > 0 ,
(1.3)
k=1
f0 (s)
= 1−
1
9
− 4 + ...
2s2
8s
as s → +∞ ,
∗ Mathematics Subject Classifications: 35P15, 35K55, 35Q55.
Key words: Ginzburg-Landau equation, spectrum, vortex dynamics, superfluid.
c
2000
Southwest Texas State University and University of North Texas.
Submitted May 1, 2000. Published June 9, 2000.
1
(1.4)
2
Ginzburg-Landau equation
EJDE–2000/42
where A0 and the P2k (A0 ) are constants with A0 positive. We consider a small
perturbation about U1 of the form
u(x) = U1 (x) + v(x),
where v is a smooth function, v(x) = 0 for x ∈ ∂B and B = B1 (0) is the unit
disk in R2 centered at the origin. Then (1.1) and (1.2) imply
L v + N (v) = 0
v=0
for x ∈ B ,
for x ∈ ∂B ,
where
L v = −∆v − 12 (1 − |U1 |2 )v + 22 (U1 · v)U1 ,
U1 · v = 12 (U¯1 v + U1 v̄),
(1.5)
N (v) = 1 [2(U1 · v)v + |v|2 U1 ] + |v|2 v .
Here the operator L : H01 (B; C) ∩ H 2 (B; C) → L2 (B; C) is the linearized operator of (1.1) about U1 .
In this paper, we find estimates for the eigenvalues and the first eigenfunction
of the operator L . Since L is self-adjoint, then all eigenvalues of L must be
real. Hence the eigenvalue problem becomes
L w = λw,
λ ∈ R,
w ∈ H01 (B; C) ∩ H 2 (B; C).
(1.6)
Lieb and Loss [15] proved that the first eigenvalue on (1.6) is nonnegative. Later,
Mironescu [21] showed that
Theorem A:
The first eigenvalue of L is positive.
In [19], we find quantitative estimates for (1.6), such as the following.
Theorem B:
Let V1 = {a(r) + b(r)e2iθ ∈ H01 (B; C) ∩ H 2 (B; C)}. Then
(i) There exist positive constants c1 , 1 independent of such that
hL w , wi ≥ c1 > 0
for w ∈ V1⊥ , kwkL2 (B) = 1 , 0 < ≤ 1 ,
where V1⊥ = {w ∈ H01 (B; C) ∩ H 2 (B; C) : hw, vi = 0 , ∀v ∈ V1 } and h·, ·i is
the inner product in L2 (B),
(ii) 0 < λ1 → 0 as → 0+.
(iii) The first eigenfunction corresponding to the first eigenvalue λ1 has the
form a (r) + b (r)e2iθ such that a ≥ b ≥ 0 for 0 ≤ r ≤ 1.
The proof of Theorem B(iii) can be found in [21]. Note that Theorem B estimates many eigenvalues but not all eigenvalues. In addition, the multiplicity of
the first eigenvalue λ1 is still unknown. Furthermore, there is no estimate about
the first eigenfunction corresponding to the first eigenvalue λ1 . We improve
Theorem B by our main results as follows.
EJDE–2000/42
Tai-Chia Lin
3
Theorem I:
(i) The eigenspace of the eigenvalue λ1 is only two-dimensional which is spanned
by two eigenfunctions a (r) + b (r)e2iθ and ia (r) − ib (r)e2iθ , where a (r)
and b (r) are real-valued.
(ii) There exist positive constants c1 , 1 independent of such that the second
eigenvalue λ2 satisfies λ2 () ≥ c1 for 0 < ≤ 1 .
(iii) 0 < λ1 = O((log 1 )−1 ) as → 0+.
¿From (1.5), L w = λ1 w, λ1 ∈ R, w ∈ V1 if and only if w = α(a(r) +
b(r)e2iθ ) + β(ic(r) + id(r)e2iθ ), for all α, β ∈ R, where a, b, c, d are real-valued
constants. However, by (1.5), the eigenfunction in V1 cannot have the form
a(r) + ib(r)e2iθ or ia(r) + b(r)e2iθ , where a, b are nonzero real-valued. Moreover,
by (1.5), w = a(r)+b(r)e2iθ ∈ V1 , where a, b are real-valued satisfies L w = λ1 w
if and only if w̃ = ia(r) − ib(r)e2iθ ∈ V1 also satisfies L w̃ = λ1 w̃. Hence the
eigenfunctions of L in V1 can be generated by all functions with the specific
forms a(r) + b(r)e2iθ and ia(r) − ib(r)e2iθ , where a, b are real. Therefore the
first eigenfunction has the specific form in Theorem I(i).
Now we introduce the stretched variable X = x/. Then we transform the
operator L into another linear operator L̃ defined by
L̃ ṽ(X) = −∆X ṽ − (1 − |Ψ0 |2 )ṽ + 2(Ψ0 · ṽ)Ψ0
(1.7)
for X ∈ B1/ (0), ṽ = ṽ(X) ∈ H01 (B1/ (0)) ∩ H 2 (B1/ (0)), where Ψ0 (X) =
2
2
f0 (s)eiθ , s = |X|, θ = arg X and ∆ = ∂X
+ ∂X
. Then L̃ ṽ = 2 λ1 ṽ if and
1
2
only if L v = λ1 v, where ṽ(X) = v(X) for X ∈ B1/ (0). Hence Theorem I(i)
implies that L̃ has the first eigenvalue 2 λ1 and the associated eigenfunctions
ẽ1 (s, θ) = ã (s) + b̃ (s)e2iθ ,
ẽ2 (s, θ) = iã (s) − ib̃ (s)e2iθ ,
where ã (s) = a (s) and b̃ (s) = b (s). We may assume that kẽj kL2 = 1 , j =
1, 2. Here k · kL2 denotes the L2 norm on B1/ (0). The estimates of ẽj ’s are
given as follows.
1
Proposition I: Let w̃j = k∂X Ψ
∂Xj Ψ0 , j = 1, 2. Assume that hw̃j , ẽj i >
0 kL 2
j
0 for j = 1, 2. Then the eigenfunctions ẽj satisfy
ẽj = w̃j + νj ,
and
1
kνj , kL2 = O((log )−1/2 )
as → + .
Theorem I and Proposition I are important tools for analyzing the vortex
solution of the Ginzburg-Landau equation
2 ∆u + (1 − |u|2 )u = 0
u = g on ∂Ω ,
in Ω ,
(1.8)
4
Ginzburg-Landau equation
EJDE–2000/42
where Ω is a bounded smooth domain, u : Ω → C is the solution and g : ∂Ω → S 1
is smooth with degree d ≥ 1. Some important results on the vortex solution of
(1.8) are presented in [1]; however, the structure of all vortex solutions in (1.8) is
still unknown. Basically, (1.8) is one of singular perturbation problems for which
by the Liapunov-Schmidt method, it is quite possible to obtain a smooth solution
of (1.8) with d degree-one vortices. One may use the symmetric vortex solution
U1 to set up an approximated solution with d degree-one vortices. However,
the spectrum of the linearized operator L is essential in the Liapunov-Schmidt
method. Therefore Theorem I and Proposition I become important for studying
the vortex solutions in (1.8).
This paper is organized as follows. In Section 2, we prove Theorem I and
Proposition I. In Section 3 and 4, we give a partial proof for the dynamics
of vortices in nonlinear heat and Schrödinger equations. This is an another
application of Theorem I and Proposition I.
2
Proof of Theorem I and Proposition I
¿From Theorem B(iii), we assume that w = a1, (r) + b1, (r)e2iθ is the first
eigenfunction of L , where a1, , b1, are real. Then L w = λ1 w becomes the
system of ordinary differential equations as follows.
−a001, − 1r a01, =
−b001, −
1 0
r b1,
+
1
2 (1
4
r 2 b1,
=
1 2
2 f b1, + λ1 a1, for
2f2 )b1, − 12 f2 a1, + λ1 b1,
− 2f2 )a1, −
r ∈ (0, 1) , (2.1)
1
2 (1
for r ∈ (0, 1) ,
−
a1, (1) = b1, (1) = 0 ,
where f (r) = f0 ( r ). ¿From [20], a1, (r), b1, (r) are real analytic for 0 ≤ r ≤ 1.
Hence
a1, (r) =
∞
X
αk rk ,
b1, (r) =
k=0
∞
X
βk r k ,
for 0 ≤ r ≤ 1 ,
(2.2)
k=0
where αk , βk ∈ R are constants. By (1.3), we have
f2 (r) = Ã r2 +
∞
X
P̃2k+2 (Ã )r2k+2
(2.3)
k=1
for 0 ≤ r ≤ 1, where à = (A0 /)2 > 0 and P̃2k+2 (à )’s are constants depending
on à .
Taking (2.2) and (2.3) into (2.1), and comparing the coefficients of rk ’s, we
obtain that
α2k+1
α2k+2
= β2k+1 = 0 ,
1
Ã
2 + λ1
= − α2k +
(2α2k−2 + β2k−2 )
4(k + 1)2
4(k + 1)2
EJDE–2000/42
Tai-Chia Lin
+
5
k−1
X
1
P̃2l+2 (Ã )(2α2(k−l−1) + β2(k−l−1) ) ,
4(k + 1)2
l=1
β2k+2
=
1
2
−
+ λ1
Ã
β2k +
(2β2k−2 + α2k−2 )
4k(k + 2)
4k(k + 2)
+
X
1
P̃2l+2 (Ã )(2β2(k−l−1) + α2(k−l−1) ) .
4k(k + 2)
k−1
l=1
Hence by induction, we have
a1, (r)
b1, (r)
=
=
α0
α0
∞
X
k=0
∞
X
C2k r2k + β2
E2k r2k + β2
k=2
∞
X
k=3
∞
X
D2k r2k ,
(2.4)
F2k r2k ,
k=1
Ã
for 0 ≤ r ≤ 1 , where C0 = 1, F2 = 1, E4 = Ã
12 , D6 = 36 and all the other
C2k , D2k , E2k and F2k ’s depend only on k, , λ1 and à .
Now we show the eigenspace of λ1 is two dimensional. By (2.4) and a1, (1) =
b1, (1) = 0, we obtain that α0 and β2 must satisfy one of the following conditions:
(1) β2 = K1 α0 and α0 is any real number, where K1 is a constant independent
of α0 .
(2) α0 = 0, β2 is any real number.
(3) both α0 and β2 are any real numbers.
Suppose (2) or (3) holds. Setting α0 = 0, β2 = 1, then
|a1, | < |b1, |
as r → 0 + .
(2.5)
However, Theorem B(iii) implies that |a1, | ≥ |b1, | for 0 ≤ r ≤ 1. This is a
contradiction with (2.5). Hence only the case (1) holds. Thus the eigenfunction
w = a1, (r) + b1, (r)e2iθ satisfies
a1, (r)
b1, (r)
=
=
∞
∞
X
X
α0 (
C2k r2k + K1
D2k r2k ) ,
k=0
∞
X
α0 (
k=2
E2k r2k + K1
k=3
∞
X
(2.6)
F2k r2k ) .
k=1
¿From (1.5), it is easy to check that L ŵ = λ1 ŵ , where ŵ = ia1, (r) −
ib1,(r)e2iθ . Therefore the eigenspace of λ1 is only two-dimensional, spanned
by a1, (r) + b1, (r)e2iθ and ia1, (r) − ib1, (r)e2iθ , and we complete the proof of
theorem I(i).
Now we prove Theorem I(ii) by contradiction. Suppose the second eigenvalue
λ2 → 0 as → 0+. Let w2, be the eigenfunction associated with λ2 . Then
6
Ginzburg-Landau equation
EJDE–2000/42
Theorem B implies that w2, ∈ V1 . Hence by (1.5), L w2, = λ2 w2, ∈ V1 and
λ2 ∈ R, we obtain that w2, must have the form w2, = a2, (r)+b2, (r)e2iθ , where
a2, , b2, are real-valued. Thus the equations L w = λ1 w and L w2, = λ2 w2,
become the systems of ordinary differntial equations as follows.
−a00j, − 1r a0j, =
−b00j, − 1r b0j, +
1
2 (1
4
r 2 bj,
=
− 2f02 ( r ))aj, −
1
2 (1
−
1 2 r
2 f0 ( )bj, + λj aj, ,
2f02 ( r ))bj, − 12 f02 ( r )aj, + λj bj,
,
aj, (1) = bj, (1) = 0 ,
(2.7)
for r ∈ (0, 1] , j = 1, 2. Now we assume that r = s, aj, (s) = aj, (s) and
bj, (s) = bj, (s) . For the notation convenience, we use the same aj , and bj ,
after the scaling r = s. Then (2.7) implies
−a00j, − 1s a0j, = (1 − 2f02 (s))aj, − f02 (s)bj, + λj 2 aj, ,
−b00j, − 1s b0j, +
4
s2 bj,
= (1 − 2f02 (s))bj, − f02 (s)aj, + λj 2 bj, ,
aj, ( 1 ) = bj, ( 1 ) = 0 ,
(2.8)
for s ∈ (0, 1 ] , j = 1, 2. By [20], we set
aj, (s) =
∞
X
αj,k sk , bj, (s) =
k=0
∞
X
βj,k sk ,
for 0 ≤ s ≤
k=0
1
, j = 1, 2 ,
where αj,k ’s and βj,k ’s are constants. ¿From (1.3), f0 (s) satisfies
f0 (s) =
A0 s +
∞
X
P2k+1 (A0 )s2k+1 ,
k=1
∞
X
f02 (s) =
Ã0 s2 +
(2.9)
P̃2k+2 (Ã0 )s2k+2 ,
(2.10)
k=1
for s > 0, where Ã0 = A20 and P̃2k+2 (Ã0 )’s are constants depending on Ã0 .
Moreover, (1.4) implies
f0 (s) =
f02 (s) =
1
9
− 4 + . . . as s → +∞ ,
2
2s
8s
1
2
1 − 2 − 4 + . . . as s → +∞ .
s
s
1−
(2.11)
(2.12)
Then by (2.8) and (2.10), we obtain a similar formula to (2.4) as follows.
aj, (s)
= αj,0
∞
X
Cj,2k s2k + βj,2
k=0
bj, (s)
= αj,0
∞
X
k=2
∞
X
Dj,2k s2k ,
(2.13)
k=3
Ej,2k s2k + βj,2
∞
X
k=1
Fj,2k s2k ,
0≤s≤
1
EJDE–2000/42
Tai-Chia Lin
7
Ã0
0
where Cj,0 = 1, Fj,2 = 1, Ej,4 = Ã
12 , Dj,6 = 36 and all the other Cj,2k ,
Dj,2k , Ej,2k and Fj,2k are depending only on , λj and Ã0 . ¿From (2.6), we
have β1,2 = K̃1 α1,0 , where K̃1 is a constant independent of α1,0 . Hence by
Theorem B(ii), (2.8), (2.13) and the standard ordinary differential equation
theorem, we have
a1, (s) = α1,0 [(a0 (s) + λ1 2 â1, (s)) + K̃1 (a1 (s) + λ1 2 ă1, (s))] ,
b1, (s) = α1,0 [(b0 (s) + λ1 2 b̂1, (s)) + K̃1 (b1 (s) + λ1 2 b̆1, (s))] ,
(2.14)
where a0 (s) = O(1), â1, (s) = O(s2 ), a1 (s) = O(s6 ), ă1, (s) = O(s8 ), b0 (s) =
O(s4 ), b̂1, (s) = O(s6 ), b1 (s) = O(s2 ), b̆1, (s) = O(s4 ) as s → 0+, and
(aj (s) , bj (s))’s are solutions of
−a00 − 1s a0 = (1 − 2f02 (s))a − f02 (s)b ,
−b00 − 1s b0 +
4
s2 b
= (1 − 2f02 (s))b − f02 (s)a for s > 0 .
(2.15)
By a2, ( 1 ) = b2, ( 1 ) = 0 and (2.13), we obtain that α2,0 and β2,2 satisfy one of
the following cases:
(a) β2,2 = K̃2 α2,0 and α2,0 is any real number, where K̃2 is a constant independent of α2,0 .
(b) α2,0 = 0, β2,2 is any real number.
(c) Both α2,0 and β2,2 are any real numbers.
For the case (a), we utilize (2.8), (2.13), λ2 → 0 as → 0+ and the standard
ordinary differential equation theorem. Then we obtain that
a2, (s) = α2,0 (a0 (s) + λ2 2 â2, (s)) + α2,0 K̃2 (a1 (s) + λ2 2 ă2, (s)) ,
b2, (s) = α2,0 (b0 (s) + λ2 2 b̂2, (s)) + α2,0 K̃2 (b1 (s) + λ2 2 b̆2, (s)) ,
(2.16)
where â2, (s) = O(s2 ), ă2, (s) = O(s8 ), b̂2, (s) = O(s6 ) and b̆2, (s) = O(s4 ) as
s → 0+. Moreover, for any M > 0 (independent of ), there exist 1 = 1 (M ) >
0 and κ = κ(M ) > 0 such that
|âj , (s)|, |b̂j , (s)|, |ăj , (s)|, |b̆j , (s)| ≤ κ
for 0 ≤ s ≤ M,
have
(2.17)
0 < ≤ 1 , j = 1, 2. Now we set α1,0 = α2,0 = 1. Then we
w2, − w = α (r) + β (r)e2iθ ,
α (r) = (K̃2 − K̃1 )a1 ( r ) + 2 [λ2 (â2, + K̃2 ă2, ) − λ1 (â1, + K̃1 ă1, )]( r ) ,
β (r) = (K̃2 − K̃1 )b1 ( r ) + 2 [λ2 (b̂2, + K̃2 b̆2, ) − λ1 (b̂1, + K̃1 b̆1, )]( r ) .
(2.18)
For (2.15), we have the following result.
8
Ginzburg-Landau equation
Lemma I
Then
EJDE–2000/42
Let (aj , bj ), j = 2, . . . , 5 , be the fundamental solutions of (2.15).
f0
f0
+ f00 , b2 (s) = f00 −
for s > 0
s
s
and the asymptotic behaviors of (aj , bj ), j = 3, 4, 5 are as follows.
a2 (s) =
√
2s −1/2
5
1
1
( − √ + O( 2 )) ,
2 16 2s
s
√
5
1
1
b3 (s) = e− 2s s−1/2 ( − √ + O( 2 )) ,
2 16 2s
s
1
1
s
s
a4 (s) = + O( ) , b4 (s) = − + O( ) ,
2
s
2
s
1 2
a5 (s), b5 (s) ≥ s as s → +∞ .
8
a3 (s) = e−
s
We will prove Lemma I in the appendix. Now we claim that
Z ∞
1 2
a (s) ds = +∞ .
s 1
1
(2.19)
We prove this by contradiction. Suppose
Z ∞
1 2
a (s) ds < +∞ .
s 1
1
Since (a1 , b1 ) is a solution of (2.15), then by Lemma I, we have
(a1 , b1 ) = α(a2 , b2 ) + β(a3 , b3 ) ,
where α, β ∈ R. Moreover, by a1 (s) ∼ s6 , b1 (s) ∼ s2 , a2 (s) ∼ 1 and b2 (s) ∼ s2
as s → 0+, then we obtain that β 6= 0 and β(a3 , b3 ) = (a1 , b1 ) − α(a2 , b2 ) is
bounded as s → 0+. Hence (a3 , b3 ) is a nontrivial bounded solution of (2.15).
From Lemma I, (a3 , b3 ) decays exponentially to zero as s → +∞. Thus (a3 , b3 )
is an eigenfunction in L2 (R2 ) × L2 (R2 ) of L associated with the eigenvalue 0,
where L is the linear operator defined by
a00 + 1s a0 + (1 − 2f02 )a − f02 b
L(a, b) =
,
b00 + 1s b0 − s42 b + (1 − 2f02 )b − f02 a
for a = a(s), b = b(s), s > 0. This is a contradiction with Proposition 5.4 in
[24]. Therefore we obtain (2.19).
Now we use (2.19) to complete the proof of Theorem I(ii). By (2.18) and
Theorem I(i) in [19], we have
1
hL (w2, − w ) , w2, − w i
2π
Z 1
4
r
r
r[(α0 )2 + (β0 )2 ] + β2 − 2 (1 − f02 ( ))(α2 + β2 ) dr
=
r
0
EJDE–2000/42
Tai-Chia Lin
Z
1
+
0
Z
1
=
0
Z
r 2 r
f ( )(α + β )2 dr
2 0 1
r
r
r(α0 )2 + α2 − 2 (1 − f02 ( ))α2 dr
r
1
+
0
Z
9
1
r
r
r(β0 )2 + β2 − 2 (1 − f02 ( ))β2 dr
r
1
r
r
3 2 1 2
β − α + 2 f02 ( )(α + β )2 dr
r
0 r
Z 1
r
r
c
3 2 1 2
kw2, − w k2L2 +
β − α + 2 f02 ( )(α + β )2 dr
2π
r
r
0
c
2
2
(kw2, kL2 + kw kL2 )
2π
Z 1/
1
3 2
β (s) − α2 (s) + sf02 (s)(α + β )2 (s) ds ,
+
s
s
0
+
≥
=
where c > 0 is a constant independent of . Then
1
2π hL (w2, − w ), w2, − w iR
1/
c
≥ 2π
(kw2, k2L2 + kw k2L2 ) + 0 3s β2 (s)
− 1s α2 (s) + sf02 (s)(α + β )2 (s) ds ,
(2.20)
Setting β = τ α , then we obtain
3
1
3 2 1 2
β − α + sf02 (s)(α + β )2 = [ τ 2 − + sf02 (s)(1 + τ )2 ]α2 .
s s
s
s
It is easy to check that
3 2 1
τ − + sf02 (s)(1 + τ )2 ≥ H0 (s) for τ ∈ R , s > 0 ,
s
s
where H0 (s) =
2s2 f02 (s)−3
.
s(3+s2 f02 (s))
Hence (2.20) becomes
1
hL (w2, − w ) , w2, − w i
2π
Z 1/
c
(kw k2L2 + kw2, k2L2 ) +
≥
H0 (s)α2 (s) ds .
2π
0
(2.21)
Note that (2.12) implies that H0 (s) ≥ 1s as s → +∞. Thus by (2.19), we may
choose a large constant M > 0 independent of such that
Z M
H0 (s)a21 (s) ds > 0 and H0 (s) > 0 for s ≥ M .
(2.22)
0
Therefore by (2.21), we have
1
hL (w2, − w ) , w2, − w i
2π
10
Ginzburg-Landau equation
≥
=
c2
Z
1/
s
0
c 2
2
+c2
(a2j, + b2j, )(s) ds +
j=1
Z
M
s
0
c
+ 2
2
2
X
2
X
M
s
0
s
M
H0 (s)α2 (s) ds
(a2j, + b2j, )(s) ds
2
X
(a2j, + b2j, )(s) ds +
j=1
1/
1/
0
j=1
Z
Z
Z
EJDE–2000/42
(2.23)
Z
M
0
Z
2
X
(a2j, + b2j, )(s) ds +
1/
M
j=1
H0 (s)α2 (s) ds
H0 (s)α2 (s) ds
Furthermore, by (2.18), we have
Z M
H0 (s)α2 (s) ds
0
2
Z
M
H0 (s)a21 (s) ds
2
Z
M
+ 2(K̃2 − K̃1 )
H0 (s)a1 (s)
= (K̃2 − K̃1 )
0
0
h
i
× λ2 (â2, + K̃2 ă2, ) − λ1 (â1, + K̃1 ă1, ) (s) ds
(2.24)
Z M
H0 (s)[λ2 (â2, + K̃2 ă2, ) − λ1 (â1, + K̃1 ă1, )]2 (s) ds
+4
0
Hence by (2.14), (2.16), (2.17), (2.22), (2.24) and α1,0 = α2,0 = 1, λ1 , λ2 → 0
as → 0+, we have
Z
Z M
2
c 2 M X 2
2
s
(aj, + bj, )(s) ds +
H0 (s)α2 (s) ds > 0
(2.25)
2
0
0
j=1
as 0 < ≤ 1 , where 1 > 0 is a small constant. Thus by (2.22), (2.23) and
(2.25), we obtain
Z
2
c 2 1/ X 2
1
hL (w2, − w ) , w2, − w i ≥
s
(aj, + b2j, )(s) ds
2π
2
0
j=1
=
c
(kw k2L2 + kw2, k2L2 ) ,
4π
then
c
1
hL (w2, − w ) , w2, − w i ≥
(kw k2L2 + kw2, k2L2 ) .
2π
4π
On the other hand, by the definition of w and w2, , we have
1
hL (w2, − w ) , w2, − w i
2π
=
=
≤
1
hλ2 w2, − λ1 w , w2, − w i
2π
1
(λ1 kw k2L2 + λ2 kw2, k2L2 )
2π
1
λ2 (kw k2L2 + kw2, k2L2 )
2π
(2.26)
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Tai-Chia Lin
11
then
1
1
hL (w2, − w ) , w2, − w i ≤
λ2 (kw k2L2 + kw2, k2L2 ) .
2π
2π
(2.27)
Therefore (2.26) and (2.27) imply
λ2 ≥
c
≡ c1 > 0 for 0 < ≤ 1 .
2
For cases (b) and (c), we set α2,0 = 0, β2,2 = 1 in (2.13). Then we obtain that
w2, = a2, (r) + b2, (r)e2iθ ,
a2, = a1 ( r ) + 2 λ2 ă2, ( r ) ,
b2, = b1 ( r ) + 2 λ2 b̆2, ( r ) .
(2.28)
Note that (2.28) has the similar form to (2.18). Hence we may apply the same
argument as (2.26) to derive that
c
1
hL w2, , w2, i ≥
kw2, k2L2 .
2π
4π
i.e. λ2 ≥ 2c ≡ c1 > 0. Therefore we complete the proof of Theorem I(ii).
For the proof of Theorem I(iii), we define the approximate eigenfunction
Ũ (s, θ) as follows.
Ũ (s, θ) = a(s) + b(s)e2iθ ,
1
0
if 0 < s ≤ R ,
s f0 (s) + f0 (s)
a(s) =
3
if R ≤ s ≤ 1 ,
B (1 − s)
0
f0 (s) − 1s f0 (s) if 0 < s ≤ R ,
b(s) =
if R ≤ s ≤ 1 ,
D (1 − s)3
where R = 1 (1 − N −1), N > 0 is a large constant independent of , and B , D
are defined by
B (1 − R )3 = R1 f0 (R ) + f00 (R ) ,
(2.29)
D (1 − R )3 = f00 (R ) − R1 f0 (R ) .
Let
U (s , θ) = C−1 Ũ (s , θ) ,
C = kŨ kL2 .
Then it is easy to check that U ∈ H01 (B1/ ) and L̃ U = 0 for 0 < s < R . In
addition, C = (4π log 1 + O(1))1/2 . Hence by (1.2), (2.11) and integration by
parts, we have
R
hL̃ U , U i = B1/ |∇X U |2 − (1 − f02 )|U |2 + 2(Ψ0 · U )2 dX
R 1/
= 2πC−2 0 s[(a0 )2 + (b0 )2 ] + 4s b2
(2.30)
−s(1 − f02 )(a2 + b2 ) + sf02 (a + b)2 ds
= O(2 (log 1 )−1 ) ,
12
Ginzburg-Landau equation
EJDE–2000/42
where h·, ·i is the product of H −1 (B1/ ) and H01 (B1/ ). Moreover,
1
kw̃1 − U kL2 = O((log )−1/2 ) .
By the standard scaling method, we obtain
(2.31)
λ1 2 ≤ hL̃ U , U i .
Hence by (2.30), we have λ1 = O((log 1 )−1 ) as → 0+. Thus we complete the
proof of theorem I(iii).
Now we prove Proposition I. By the definition of U and ẽ1 , there exist
γ ∈ R and ν ∈ H01 (B1/ (0)) such that
U = γ ẽ1 + ν ,
ν ⊥ẽ1 in L2 (B1/ (0))
and γ = hU , ẽ1 i.
(2.32)
Hence by Theorem I(ii), we obtain
hL̃ U , U i =
=
≥
then
hL̃ (γ ẽ1 + ν ) , γ ẽ1 + ν i
γ2 2 λ1 + hL̃ ν , ν i
c1 2 kν k2L2 ,
hL̃ U , U i ≥ c1 2 kν k2L2 ,
(2.33)
where c1 > 0 is a constant independent of . Thus by (2.30) and (2.33), we have
1
kν kL2 = O((log )−1/2 ) .
(2.34)
¿From (2.32) and the definition of U , we obtain
kU k2L2 = γ2 + kν k2L2 = 1 .
Therefore by (2.34), we have γ2 = 1 + O((log 1 )−1 ). Since hw̃1 , ẽ1 i > 0 and by
(2.32), then γ > 0. Hence
1
γ = (1 + O((log )−1 ))1/2 .
By (2.31) and (2.35), we obtain
1
1
k U − w̃1 kL2 = O((log )−1/2 )
γ
(2.35)
(2.36)
From (2.32) and (2.35), we have
ẽ1 =
where ν1 , =
1
γ (U
1
(U − ν ) = w̃1 + ν1, ,
γ
− ν ) − w̃1 . By (2.34), (2.35) and (2.36), we obtain that
1
kν1 , kL2 = O((log )−1/2 ) .
Similarly, we have
1
ẽ2 = w̃2 + ν2 , , kν2 , kL2 = O((log )−1/2 ) .
Therefore we complete the proof of Proposition I.
EJDE–2000/42
3
Tai-Chia Lin
13
Vortex dynamics in nonlinear heat equations
In this Section, we consider the system of nonlinear heat equations
ut = ∆u + 12 u(1 − |u|2 ) for x ∈ R2 , t > 0,
u|t=0 = u0 (x) for x ∈ R2 .
(3.1)
The equation (3.1) is the simplified Ginzburg-Landau equation which is from
the Ginzburg-Landau theory of superconductivity. As is small, the GinzburgLandau theory predicted the existence of vortex state. The vortex state consists
of many normal filaments embedded in a superconducting matrix. Each of these
filaments carries with it a quantized amount of magnetic flux, and is circled by
a vortex of superconducting current. The vortex structures are set in motion by
a variety of mechanism, including thermal fluctuations and applied voltages and
currents. Unfortunately, such vortex motion in an applied magnetic field induces
an effective resistance in the material, and thus a loss of superconductivity.
Therefore, it is crucial to understand the dynamics of these vortices in order to
pin vortices at a fixed location, i.e. prevent their motion. Hereafter, vortices are
so called point vortices which come from the cross section of vortex filaments.
For the dynamics of vortices, E [5] used the formal asymptotic analysis
to derive the dynamic laws of vortices. In this Section, we will give a more
general proof of the dynamics of vortices. The idea of the proof is to use the
generalized asymptotic expansion (3.5), Theorem I and Proposition I. Assume
that x = q (t) ∈ R2 is the vortex trajectory, q is smooth in t, q (0) = 0,
and B1 (q (t)) is the vortex core which moves along with the vortex trajectory
x = q (t). Here we assume that the solution u has only one vortex center at
q (t). Now we focus on the vortex core B1 (q (t)) and consider the following
system of equations
ut = ∆u + 12 u(1 − |u|2 )
u|t=0 = u0 (x)
for x ∈ B1 (q (t)), t > 0,
for x ∈ B1 (0) .
(3.2)
Assume that
x − q (t)
, Ψ(X , t , ) = u(x , t , )
for x ∈ B1 (q (t)) , i.e.X ∈ B1/ (0) , t ≥ 0 . Then we have
X=
2 Ψt = q̇ · ∇X Ψ + ∆X Ψ + Ψ(1 − |Ψ|2 ) for X ∈ B1/ (0), t > 0,
Ψ|t=0 = u0 for X ∈ B1/ (0) ,
where Ψt =
∂
∂t Ψ(·, t, ).
(3.3)
(3.4)
We take an expansion form of Ψ as follows.
Ψ(X, t, ) = Ψ0 (X)eiH + Ψ1 (X, t, )eiH ,
(3.5)
where Ψ0 (X) = f0 (s)eiφ , s = |X|, φ = arg(X). Note that (3.6) is a more
general form than the inner expansion form in [22] and [5]. Here we assume
that H = H(x, t, ) is a smooth real-valued function and satisfies
∆x H = 0 , |∇x H| , |Ht | , |∇x Ht | ≤ K in B1 (q (t)) ,
(3.6)
14
Ginzburg-Landau equation
EJDE–2000/42
∂
H and K > 0 is a constant. We will derive the governing equation
where Ht = ∂t
of the vortex trajectory x = q (t) as follows.
q̇
=
=
1
[2J∇x H(q , t , ) − c J∇x Ht (q , t , ) + o (1)]
1
[−2∇x H̃(q , t , ) + c ∇x H̃t (q , t , ) + o (1)] ,
log 1 +O(1)
log
1
+O(1)
(3.7)
0 1 , and H, H̃ are harmonic conjugate.
0
−1 0
Here we denote o (1) as a small quantity, independent of time t, and tending to
zero as → 0+. ¿From (3.7), it is remarkable that the velocity q̇ of the vortex
trajectory x = q (t) can be nonzero and of order O((log 1 )−1 ) if the function H
is nonconstant. We require that Ψ1 satisfies the ”small” perturbation condition
as follows.
k∇X Ψ1 kL2 (B 1 ) ≤ K−β , 0 < β < 1 ,
where c = 2
R 1/
sf02 (s) ds, J =
kΨ1 kL6 (B 1 ) ≤ K−γ ,
0<γ<
kΨ1 ,t kL2 (B 1 ) ≤ K−δ ,
1
3
,
(3.8)
0 < δ < 2,
|
R
|hΨ1 , ẽj i| = o (−2 (log 1 )1/2 ) , j = 1, 2 ,
P∞ PJ(k) 2
k=2
j=1 λk |hΨ1 , ẽj ,k i| = o (1) ,
Ψ1 · ∂~n ẽj | = o ((log 1 )−1/2 ) , j = 1, 2 ,
∂B1/
P∞ PJ(k) R
n ẽj ,k | = o (1) ,
k=2
j=1 | ∂B1/ Ψ1 · ∂~
(3.9)
(3.10)
where J(k) is the multiplicity of the eigenvalue λk , K > 0 is a constant and
∂~n is the normal derivative. Here ẽj ’s and ẽj ,k ’s are the eigenfunctions in
Ṽ1 = {a(s) + b(s)e2iφ ∈ H01 (B1/ ; C)} of L̃ corresponding to the eigenvalues
2 λ1 and 2 λk ’s respectively. We require them to have unit norm in L2 (B1/ ).
In addition h· , ·i is the inner product in L2 . Note that the upper bound of
(3.8) and the first term of (3.9) tend to infinity as goes to zero. ¿From [9],
the equation (3.2) is well posed. Then Ψ1 is smooth in both space and time
variables. Hence (3.8), (3.9) and (3.10) can be fulfilled in a short time when
Ψ1 |t=0 satisfies the following assumption.
Assumption I: Ψ1 = Ψ1 (X, t, ) satisfies the following:
(1) Ψ1 (·, 0, ) has sufficiently small C 1 norm on the vortex core {X : |X| ≤ 2 }
at t = 0
(2) k∂t Ψ1 (·, 0, )kL2 (B 2 ) = O(−δ ), 0 < δ < 2.
By the suitable choice of initial data u0 , we obtain Assumption I(1). Assumption I(2) preserves the vortex structure on the vortex core when the associated
vortex point begins to move at the time t = 0. Note that the upper bound of
Assumption I(2) is ∼ −δ , with 0 < δ < 2 which tends to infinity as goes to
zero. We observe that Assumption I is a local perturbation condition on the
vortex core which is different from the global assumption in [16] and [17].
EJDE–2000/42
Tai-Chia Lin
15
In Neu [22] and E [5]’s works, they used a specific asymptotic expansion formula and a pointwise matching condition to derive the dynamic laws of vortices.
Basically, they require some pointwise matching condition on the boundary of
vortex cores to derive the dynamic laws of vortices. However, (3.8), (3.9) and
(3.10) are not pointwise. This is a kind of generalization for the results of Neu
and E.
Now we prove (3.7) as follows. By (3.5) and (3.6), (3.4) becomes
q̇ · (∇X Ψ0 + ∇X Ψ1 ) − iΨ0 Ht − i2 Ψ1 Ht − 2 Ψ1,t
(3.11)
2
Ψ0 |∇x H| − 2i(∇X Ψ0 · ∇x H) + L̃ Ψ1 + N̂ (Ψ1 )
=
for X ∈ B1/ (0) and t > 0, where
−L̃ Ψ1 =
N̂ (Ψ1 ) =
and Ψ1,t =
w̃j =
∆X Ψ1 + (1 − |Ψ0 |2 )Ψ1 − 2(Ψ0 · Ψ1 )Ψ0 ,
2 Ψ1 |∇x H|2 − 2i(∇X Ψ1 · ∇x H) + 2 |Ψ1 |2 Ψ1
+[2(Ψ0 · Ψ1 )Ψ1 + |Ψ1 |2 Ψ0 ] ,
∂
∂t Ψ1 (·, t, ).
1
∂X Ψ0 ,
Γj j
(3.12)
Assume that
Γj = k∂Xj Ψ0 kL2 = (π log
1
+ O(1))1/2 ,
j = 1, 2 . (3.13)
Hereafter, we use Xj to denote the components of X = (X1 , X2 ). Then (3.11)
implies
q̇,1 (∂X1 Ψ0 + ∂X1 Ψ1 ) + q̇,2 (∂X2 Ψ0 + ∂X2 Ψ1 ) − iΨ0 Ht − i2 Ψ1 Ht − 2 Ψ1,t
= Ψ0 |∇x H|2 − 2i(∂X1 Ψ0 ∂x1 H + ∂X2 Ψ0 ∂x2 H) + L̃ Ψ1 + N̂ (Ψ1 )
(3.14)
for X ∈ B1/ (0) and t > 0. Making the inner product with (3.14) and w̃j ,
j = 1, 2, we have
q̇,1 [h∂X1 Ψ0 , w̃j i + h∂X1 Ψ1 , w̃j i] + q̇,2 [h∂X2 Ψ0 , w̃j i + h∂X2 Ψ1 , w̃j i]
= 2 (hΨ1,t , w̃j i + hiΨ1 Ht , w̃j i) + hN̂ (Ψ1 ), w̃j i + Γj + hL̃ Ψ1 , w̃j i ,
(3.15)
for j = 1, 2, where h·, ·i is the inner product in L2 (B1/ (0); C), and
Γj
=
−2hi∂X1 Ψ0 ∂x1 H , w̃j i − 2hi∂X2 Ψ0 ∂x2 H , w̃j i
+hΨ0 |∇x H|2 , w̃j i + hiΨ0 Ht , w̃j i .
(3.16)
Since the eigenfunctions ẽj , ẽj ,k ’s dense in Λ1 = {w(X) = a(s) + b(s)e2iφ ∈
L2 (B1/ (0); C)} and w̃l ∈ Λ1 , l = 1, 2, then
w̃l =
2
X
j=1
hw̃l , ẽj iẽj +
∞ J(k)
X
X
hw̃l , ẽj ,k iẽj ,k .
(3.17)
k=1 j=1
Using integration by parts, we have
R
hL̃ Ψ1 , ẽj i
= ∂B1/ Ψ1 · ∂~n ẽj + 2 λ1 hΨ1 , ẽj i ,
R
hL̃ Ψ1 , ẽj ,k i = ∂B1/ Ψ1 · ∂~n ẽj ,k + 2 λk hΨ1 , ẽj ,k i .
(3.18)
16
Ginzburg-Landau equation
EJDE–2000/42
Hence by Proposition I, (3.9), (3.10), (3.17) and (3.18), we obtain
R
P
hL̃ Ψ1 , w̃l i = 2j=1 (2 λ1 hΨ1 , ẽj i + ∂B1/ Ψ1 · ∂~n ẽj )hẽj , w̃l i
R
P∞ PJ(k)
+ k=2 j=1 (2 λk hΨ1 , ẽj ,k i + ∂B1/ Ψ1 · ∂~n ẽj ,k )hẽj ,k , w̃l i
= o ((log 1 )−1/2 ) , l = 1, 2 .
(3.19)
It is easy to check that
∂X1 Ψ0
=
∂X2 Ψ0
=
1 1
2 ( s f0
i 1
2 ( s f0
+ f00 ) + 12 (f00 − 1s f0 )e2iφ ,
+ f00 ) + 2i ( 1s f0 − f00 )e2iφ .
(3.20)
Hence h∂X1 Ψ0 , w̃2 i = h∂X2 Ψ0 , w̃1 i = 0 . Thus (3.15) and (3.19) imply that
q̇,1 (α∗ + α1 ) + q̇,2 β1 =
q̇,1 β2 + q̇,2 (α∗ + α2 ) =
γ1 ,
γ2 ,
(3.21)
where
α∗ = Γj = k∂Xj Ψ0 kL2 , αj = h∂Xj Ψ1 , w̃j i ,
β1 = h∂X2 Ψ1 , w̃1 i, β2 = h∂X1 Ψ1 , w̃2 i, ηj = hN̂ (Ψ1 ) , w̃j i ,
and
1
γj = Γj + 2 (hΨ1,t , w̃j i + hiΨ1 Ht , w̃j i) + ηj + o ((log )−1/2 ) ,
for j = 1, 2. By (3.6), (3.8) and Hölder inequality, we have
|αj | ≤ K1−β , |βj | ≤ K1−β , 0 < β < 1 ,
|ηj | = o ((log 1 )−1/2 ) , γj = Γj + o ((log 1 )−1/2 ) , j = 1, 2 .
(3.22)
Moreover, (3.21) implies
q̇,1 =
1
[(α∗ + α2 )γ1 − β1 γ2 ] ,
γ
q̇,2 =
1
[(α∗ + α1 )γ2 − β2 γ1 ] ,
γ
(3.23)
where γ = α2∗ + α∗ (α1 + α2 ) + 2 (α1 α2 − β1 β2 ). Furthermore by (3.6), (3.20),
and the mean-value theorem of harmonic functions, we have
hi∂Xj Ψ0 ∂xj H , w̃j i = 0 , j = 1, 2 ,
hi∂X1 Ψ0 ∂x1 H , w̃2 i = Γπ2 ∂x1 H(q , t , )f02 ( 1 ) ,
hi∂X2 Ψ0 ∂x2 H , w̃1 i = − Γπ1 ∂x2 H(q , t, )f02 ( 1 ) ,
R 1/
2
,
C
=
2πK
sup
sf0 f00 ds > 0 ,
|hΨ0 |∇x H|2 , w̃j i| ≤ ΓC
0<<1
0
j
R
1/
hiΨ0 Ht , w̃1 i = − Γπ1 (2 0 sf02 ds)∂x2 Ht (q , t , ) ,
R
1/
hiΨ0 Ht , w̃2 i = Γπ2 (2 0 sf02 ds)∂x1 Ht (q , t , ) .
(3.24)
Hence by (1.3), (1.4), (3.16), and (3.24), we obtain
Γ1
Γ2
=
=
π
1 −1/2
),
Γ1 (2∂x2 H(q , t , ) − c ∂x2 Ht (q , t , )) + O((log )
π
− Γ2 (2∂x1 H(q , t , ) − c ∂x1 Ht (q , t , )) + O((log 1 )−1/2 ) ,
(3.25)
EJDE–2000/42
Tai-Chia Lin
17
R 1/
where c = 2 0 sf02 ds > 0. Note that by (1.3) and (1.4), c ∼ 1/2 as → 0+ .
Thus by (3.13), (3.22), (3.23), and (3.25), we complete the proof of (3.7).
For the motion of d-vortices, we restrict (3.1) on the vortex cores B1 (qj )’s
and we consider the following system of equations.
ut = ∆u + 12 u(1 − |u|2 ) for x ∈ B1 (qj (t)), t > 0,
u|t=0 = uj0 (x) for x ∈ B1 (qj (0)) ,
where x = qj (t) is the j-th vortex trajectory, qj is smooth in t, |qj − qk | > 2
for j 6= k, and B1 (qj (t)) is the j-th vortex core which moves along with the
j-th vortex trajectory x = qj (t), j = 1, . . . , d. Now we assume that
Xj =
x − qj (t)
,
Ψ(X j , t, ) = u(x, t, )
(3.26)
for x ∈ B1 (qj (t)) i.e. X j ∈ B1/ (0) , j = 1, . . . , d . Like (3.5), we take a similar
expansion form of Ψ on each vortex core as follows.
Ψ(X j , t, ) = Ψ0 (X j )eiHj + Ψ1 (X j , t, )eiHj for X j ∈ B1/ (0) ,
(3.27)
where
Ψ0 (X j ) = f0 (sj )eiφj , sj = |X j |, φj = arg X j .
Here we assume that
P
Hj = k6=j φk + H , ∆x H = 0 in B1 (qj (t)) ,
|∇x H| , |Ht | , |∇x Ht | ≤ K in B1 (qj (t)) .
(3.28)
By the same argument as (3.7), we obtain the equations of qj as follows.
q̇j
=
=
1
[2J∇x Hj (qj , t , ) − c J∇x ∂t Hj (qj , t , ) + o (1)] ,
1
[−2∇x H̃j (qj , t , ) + c ∇x ∂t H̃j (qj , t , ) + o (1)] ,
log 1 +O(1)
log
1
+O(1)
(3.29)
where Hj , H̃j0 s are harmonic conjugates. Here we require that Ψ1 satisfies the
”small” perturbation conditions on each vortex core |X j | ≤ 1 , j = 1, . . . , d as
follows.
k∇X j Ψ1 kL2 (B 1 ) ≤ K−β , 0 < β < 1 ,
kΨ1 kL6 (B 1 ) ≤ K−γ ,
kΨ1 ,t kL2 (B 1 ) ≤ K
−δ
0<γ<
,
1
3
,
(3.30)
0 < δ < 2,
|
R
|hΨ1 , ẽm i| = o (−2 (log 1 )1/2 ) , m = 1, 2 ,
P∞ PJ(k) 2
k=2
j=1 λk |hΨ1 , ẽm ,k i| = o (1) ,
Ψ · ∂~n ẽm | = o ((log 1 )−1/2 ) ,
∂B1/ 1
P∞ PJ(k) R
n ẽm ,k | =
k=2
m=1 | ∂B1/ Ψ1 · ∂~
(3.31)
m = 1, 2 ,
o (1) ,
(3.32)
18
Ginzburg-Landau equation
EJDE–2000/42
where J(k) is the mutliplicity of the eigenvalue λk and K > 0 is a constant. In
particular, suppose H ≡ 0. Then (3.28) and (3.29) imply that
q̇j =
−2
∇x H̃j (qj , t , ) + o (1) .
log 1
(3.33)
Note that the equation (3.33) is consistent with the governing equation in [5]
and [18].
4
Vortex dynamics in nonlinear Schrödinger equations
The other application of Theorem I and Proposition I is for the dynamics
of vortices in the nonlinear Schrödinger equation. The system of nonlinear
Schrödinger equations is as follows.
−iut = ∆u + 12 u(1 − |u|2 ) for x ∈ R2 , t > 0,
u|t=0 = u0 (x) for x ∈ R2 .
(4.1)
The equation (4.1) is a fundamental equation for understanding superfluids, see
Ginzburg and Pitaevskii [8], Landau and Lifschitz [14], Donnelly [4], Frisch,
Pomeau and Rica [7], Josserand and Pomeau [13], and many others.
For the vortex dynamics in a superfluid, we prove rigorously the asymptotic
motion equation of a vortex from a solution of (4.1) with some specific conditions. The method of our proof is more generalized than the formal asymptotic
analysis in the dynamics of fluid dynamic vortices (cf. [22]). Fortunately, the
asymptotic motion equation has the same leading order term (up to a time
scaling) as Neu [22]’s result.
In [2], superfluid 4 He has a much larger Reynolds number than the Reynolds
numbers of water and air. Moreover, the helium liquids have kinematic viscosities which are much smaller than those of water. It is well known that the
high Reynolds number may cause the turbulent flow (cf. [6]). However, up to
now, it is often tacitly assumed that the laminar analysis of [5] and [22] may
carry over to turbulent vortex cores, but no theoretical corroboration of that
assumption had been available. In this paper, we provide a more generalized
and rigorous argument to derive the vortex dynamics in a superfluid. In particular, the constraints imposed concern only global norms of the perturbations
from the leading order steady state structure. Hence certain classes of highly
unsteady fluctuations are allowed for the constraints. This is an important step
to investigate the dynamics of fluid dynamic vortices with turbulent cores.
Assume that x = q (t) = (q,1 (t), q,2 (t)) is the vortex trajectory and q (0) =
0. In addition, we assume that in the vortex core B1 (q (t)), the classical solution
u(x, t) has only one vortex center at q (t). Now we focus on the vortex core
B1 (q (t)) and we consider the following equations
−iut = ∆u + 12 u(1 − |u|2 ) for x ∈ B1 (q (t)), t > 0,
u|t=0 = u0 (x) for x ∈ B1 (0),
(4.2)
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Tai-Chia Lin
19
Like (3.3) and (3.5), we take the same expansion form of the solution u as
follows.
u(x, t, ) = Ψ(X, t, ) = Ψ0 (X)eiH + Ψ1 (X, t, )eiH ,
(4.3)
where X = x−q (t) , Ψ0 (X) = f0 (s)eiφ , s = |X|, φ = arg(X), and H satisfies
(3.6).
We use Theorem I and Proposition I to derive the equation
q˙
=
=
2∇x H(q , t , ) + o (1) ,
2J∇x H̃(q , t , ) + o (1) ,
(4.4)
provided that the same ”small” perturbation conditions (3.8), (3.9) and (3.10)
hold. In [23] p. 17, we learned that the equation (4.1) is well posed. Then Ψ1 is
smooth in both space and time variables. Hence (3.8), (3.9) and (3.10) can be
fulfilled in a short time as Ψ1 |t=0 satisfies Assumption I in Section 3. By (4.3)
and (3.6), (4.2) becomes
−iq̇ · (∇X Ψ0 + ∇X Ψ1 ) − Ψ0 Ht − 2 Ψ1 Ht + i2 Ψ1,t
=
Ψ0 |∇x H|2 − 2i(∇X Ψ0 · ∇x H) + L̃ Ψ1 + N̂ (Ψ1 )
(4.5)
for X ∈ B1/ (0) and t > 0, where L̃ and N̂ are defined in (3.12). Making the
inner product with (4.5) and w̃j , j = 1, 2 and using (3.19), we have
−q̇,1 [hi∂X1 Ψ0 , w̃j i + hi∂X1 Ψ1 , w̃j i]
−q̇,2 [hi∂X2 Ψ0 , w̃j i + hi∂X2 Ψ1 , w̃j i]
=
(4.6)
1
Γ̂j + 2 (hiΨ1,t , w̃j i + hΨ1 Ht , w̃j i) + hN̂ (Ψ1 ) , w̃j i + o ((log )−1/2 ) ,
where
Γ̂j
=
−2hi∂X1 Ψ0 ∂x1 H, w̃j i − 2hi∂X2 Ψ0 ∂x2 H , w̃j i
+hΨ0 |∇x H|2 , w̃j i + hΨ0 Ht , w̃j i .
(4.7)
By (3.13) and (3.20), we have hi∂X1 Ψ0 , w̃1 i = hi∂X2 Ψ0 , w̃2 i = 0. Hence (4.6)
becomes
q̇,1 α̂12 + q̇,2 (α̂11 + β̂1 ) = −γ1 ,
(4.8)
q̇,1 (α̂21 + β̂2 ) + q̇,2 α̂22 = −γ2
where
α̂11 = hi∂X2 Ψ0 , w̃1 i , α̂21 = hi∂X1 Ψ0 , w̃2 i , α̂j2 = hi∂Xj Ψ1 , w̃j i ,
β̂1 = hi∂X2 Ψ1 , w̃1 i , β̂2 = hi∂X1 Ψ1 , w̃2 i , ηj = hN̂ (Ψ1 ), w̃j i ,
and
1
γj = Γ̂j + 2 (hΨ1 Ht , w̃j i + hiΨ1,t , w̃j i) + ηj + o ((log )−1/2 ) .
By (3.6), (3.8) and (3.20), we have
α̂11 = −π/Γ1 , α̂21 = π/Γ2 ,
|α̂j2 | ≤ K1−β , |β̂j | ≤ K1−β , 0 < β < 1 ,
|ηj | = o ((log 1 )−1/2 ) , γj = Γ̂j + o ((log 1 )−1/2 ) , j = 1, 2 .
(4.9)
20
Ginzburg-Landau equation
EJDE–2000/42
Furthermore, (4.8) implies
1
q̇,1 (t) = − [(α̂11 + βˆ1 )γ2 − α̂22 γ1 ] ,
γ̂
1
q̇,2 (t) = − [(α̂21 + βˆ2 )γ1 − α̂12 γ2 ] ,
γ̂
(4.10)
where γ̂ = (α̂11 + βˆ1 )(α̂21 + βˆ2 ) − 2 α̂12 α̂22 . Moreover, by (3.6), (3.20) and
the Mean Value Theorem of harmonic functions, we have
Z 1/
π2
hΨ0 Ht , w̃j i =
(
s2 f0 f00 ds)∂xj Ht (q , t , ) .
Γj 0
(4.11)
¿From (1.3) and (1.4), we obtain that
Z
1/
0
s2 f0 f00 ds = log
1
+ O(1) .
(4.12)
Hence by (4.7), (4.11), (4.12) and (3.24), we obtain
Γ̂1
=
Γ̂2
=
π
1
2
Γ1 [2∂x2 H(q , t , ) + (log + O(1))∂x1 Ht (q , t , )]
+O((log 1 )−1/2 ) ,
π
1
2
Γ2 [−2∂x1 H(q , t , ) + (log + O(1))∂x2 Ht (q , t , )]
1 −1/2
),
+O((log )
(4.13)
Thus by (3.6), (4.9), (4.10) and (4.13), we complete the proof of (4.4).
For the motion of d-vortices, we restrict (4.1) on the vortex cores B1 (qj )’s
and we consider
−iut = ∆u + 12 u(1 − |u|2 ) for x ∈ B1 (qj (t)), t > 0 ,
u|t=0 = uj0 (x) for x ∈ B1 (qj (0)) ,
(4.14)
where x = qj (t) is the j-th vortex trajectory, qj is smooth to t, |qj − qk | > 2
for j 6= k, and B1 (qj (t)) is the j-th vortex core which moves along with the
j-th vortex trajectory x = qj (t), j = 1, . . . , d . By the same argument as (4.4)
for each vortex core, we obtain the equations of qj as follows.
q̇j = 2J∇x H̃j (qj , t , ) + o (1) , j = 1, . . . , d ,
(4.15)
provided that the same “small” perturbation conditions (3.30), (3.31) and (3.32)
hold. Note that both (4.4) and (4.15) are consistent with the governing equations in [22] and [17].
Appendix: Proof of Lemma I
It is easy to check that a2 (s) =
To obtain (a3 , b3 ), we let
a(s) = e−
√
f0
s
2s −1/2
s
+ f00 , b2 (s) = f00 −
η1 (s) ,
b(s) = e−
f0
s
is a solution of (2.15).
√
2s −1/2
s
η2 (s)
EJDE–2000/42
Tai-Chia Lin
21
be the solution of (2.15). Then η1 and η2 satisfy
√
η100 − 2 2η10 + (3 + 4s12 − 2f02 )η1 − f02 η2 = 0 ,
√
15
2
2
η200 − 2 2η20 + (3 − 4s
2 − 2f0 )η2 − f0 η1 = 0 .
Let u = η1 + η2 and v = η1 − η2 . Then by (2.12) and (4.1), we have
√
u00 − 2 2u0 + s22 v + β1 (s)u = 0 ,
√
v 00 − 2 2v 0 + 2v + s22 u + β2 (s)v = 0 ,
(4.1)
(4.2)
where
β1 (s)
=
3−
β2 (s)
=
1−
7
4s2
7
4s2
5
6
4s2 + s4 + . . . ,
− 4s32 + s24 + . . .
− 3f02 =
− f02 =
as s → +∞ .
Now we transform (4.2) into the following integral equations
R∞
u(s) = 1 + s β1 (ζ)K(s, ζ)u(ζ) + ζ22 K(s, ζ)v(ζ) dζ ,
R∞
v(s) = s β2 (ζ)K̃(s, ζ)v(ζ) + ζ22 K̃(s, ζ)u(ζ) dζ ,
where
√
√
1 − e−2 2(ζ−s)
√
, K̃(s, ζ) = (s − ζ)e− 2(ζ−s) ,
K(s, ζ) =
−2 2
By the iteration method (cf. [12] p.199-209), we set
u0 (s) ≡ 1 , v0 (s) ≡ 0 ,
R∞
um+1 (s) = 1 + 0 β1 (ζ)K(s , ζ)um (ζ) + ζ22 K(s , ζ)vm (ζ) dζ ,
R∞
vm+1 (s) = 0 β2 (ζ)K̃(s , ζ)vm (ζ) + ζ22 K̃(s , ζ)um (ζ) dζ ,
for m = 0, 1, 2, . . .. Note that
R∞
K(s , ζ)ζ −m dζ = O(s1−m ) ,
s
R∞
K̃(s , ζ)ζ −m dζ = O(s−m ) ,
s
for m ≥ 2 ,
for m ≥ 2 .
Then it is easy to deduce that um , vm converge to u, v respectively as m → ∞.
Moreover, (u, v) is a solution of (4.2), and it satisfies
α̃32
5
u(s) = 1 − √ + 2 + . . . ,
s
8 2s
β̃33
1
v(s) = − 2 + 3 + . . . as s → +∞ ,
s
s
where α̃jk , β̃jk ’s are constants. Thus
η1
=
η2
=
5
α32
1
− √ + 2 + ... ,
2 16 2s
s
5
β32
1
− √ + 2 + ...
as s → +∞ ,
2 16 2s
s
22
Ginzburg-Landau equation
EJDE–2000/42
where αjk , βjk are constants. Therefore. we obtain the solution (a3 , b3 ).
To obtain (a4 , b4 ), we set
sη3 = a(s) + b(s) , sη4 = a(s) − b(s) ,
where (a, b) is the solution of (2.15). Then (η3 , η4 ) satisfies
η300 − 2η3 + 3s η30 + s22 η4 + β3 (s)η3 = 0 ,
η400 + 3s η40 + s22 η3 + β4 (s)η4 = 0 ,
where
β3 (s) = 3 − 3f02 − s12 = s22 + s64 + . . . ,
β4 (s) = 1 − f02 − s12 = s24 + . . . as s → +∞ .
This is equivalent to the following integral equations
R∞
η3 (s) = −3R s ( 1t K̃t (s, t) − t12 K̃(s, t))η3 (t) dt
∞
+ s [ t22 η4 (t) + β3 (t)η3 (t)]K̃(s, t) dt ,
R∞
η4 (s) = 1 + 12 s (t−2 − s−2 )t2 [β4 (t)η4 (t) + t22 η3 (t)] dt ,
where
K̃(s, t) = (s − t)e−
√
2(t−s)
,
√
√
K̃t (s, t) = −[ 2(s − t) + 1]e− 2(t−s) .
By the iteration method, we set
η3 ,0 (s)
η3 ,m+1 (s)
η4 ,m+1 (s)
≡ 0,
η4 ,0 (s) ≡ 1 ,
R∞ 1
= −3R s ( t K̃t (s, t) − t12 K̃(s, t))η3 ,m (t) dt
∞
+ s [ t22 η4 ,m (t) + β3 (t)η3 ,m (t)]K̃(s, t) dt ,
R∞
= 1 + 12 s (t−2 − s−2 )t2 [β4 (t)η4 ,m (t) + t22 η3 ,m+1 (t)] dt ,
for m = 0, 1, 2, . . ..
Now we claim that (η3 ,m , η4 ,m ) converges to the solution (η3 , η4 ). For
T u(s)
(u, v) = (u(s), v(s)), let T (u, v)(s) =
, where
T v(s)
R∞
T u(s) = −3R s ( 1t K̃t (s, t) − t12 K̃(s, t))u(t) dt
∞
+ s [ t22 v(t) + β3 (t)u(t)]K̃(s, t) dt ,
R∞
T v(s) = 1 + 12 s (t−2 − s−2 )t2 [β4 (t)v(t) + t22 T u(t)] dt ,
0
η3 ,m+1
=
+ T (η3 ,m , η4 ,m );
Then we have
1
η4 ,m+1
η
3 ,m
i.e.,
= (I + T + T 2 + . . . + T m )(0, 1). Hence it is easy to check that
η4 ,m
R ∞ −2
2
1−m
(s − t−2 )t2−m dt = (m−2)
,
for m ≥ 4 ,
2 −1 s
s
R∞
K̃t (s , t)t−m dt = O(s−(m+1) ) , for m ≥ 2 ,
s
R∞
−m
dt = O(s−m ) ,
for m ≥ 2 .
s K̃(s , t)t
EJDE–2000/42
Tai-Chia Lin
23
Thus we obtain that
O(s−4 )
O(s−2 )
, T 2 (0, 1) =
,
T (0, 1) =
−3
−5
O(s )
O(s )
−6 −8 O(s )
O(s )
, T 4 (0, 1) =
,... .
T 3 (0, 1) =
O(s−7 )
O(s−9 )
Thus
|T m (0, 1)| ≤ C2 S0−m
for m ≥ 1 , s ≥ S0 ,
where C2 > 0 is a constant and S0 > 0 is a large constant. Therefore (η3 ,m , η4 ,m )
converges to the solution (η3 , η4 ) as m → ∞. Moreover
η3 = O(
1
),
s2
η4 = 1 + O(
1
)
s2
as s → ∞ ,
and we obtain the solution (a4 , b4 ) = ( 12 s(η3 + η4 ) , 12 s(η3 − η4 )).
To obtain (a5 , b5 ), we use the change of variable s = et and let â(t) =
t
a(e ), b̂(t) = b(et ). Then (2.15) becomes
â00
=
e2t (2f02 (et ) − 1)â + e2t f02 (et )b̂ ,
b̂00
=
e2t f02 (et )â + [e2t (2f02 (et ) − 1) + 4]b̂ .
(4.3)
By (2.12), there exists a large constant T0 > 0 such that
2f02 (et ) − 1 > 0
for t ≥ T0 .
Let (â5 , b̂5 ) be the solution of (4.3) with initial data
(â5 , â05 , b̂5 , b̂05 )(T0 ) = (1, 1, 1, 1) .
Then (4.3) implies that â5 , b̂5 are positive and increasing for t ≥ T0 . Moreover
by (2.12) and (4.3), we have
â005 (t) ≥ e2t for t ≥ T1 ,
where T1 ≥ T0 is a large constant. Similarly b̂005 (t) ≥ e2t for t ≥ T1 . Hence
â5 (t) , b̂5 (t) ≥
1 2t
e
8
as t → +∞ .
Thus
a5 (s) = â5 (ln s), b5 (s) = b̂5 (ln s) ≥
1 2
s
8
as s → +∞ .
Therefore we complete the proof of lemma I.
Acknowledgement The author wants to express his gratitude to Professors
F. H. Lin, L. C. Evans, and J. C. Neu for their helpful discussions.
24
Ginzburg-Landau equation
EJDE–2000/42
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Tai-Chia Lin
Department of Mathematics
National Chung-Cheng University
Chia-Yi, Taiwan, ROC
e-mail: tclin@math.ccu.edu.tw